An athlete throws a basketball upward from the ground, giving it speed 10.8 m/s at an angle of 64.0° above the horizontal.
(a) What is the acceleration of the basketball at the highest point in its trajectory?
(b) On its way down, the basketball hits the rim of the basket, 3.05 m above the floor. It bounces straight up with one-third the speed with which it hit the rim. What height above the floor does the basketball reach on this bounce?
To answer these questions, we need to analyze the motion of the basketball at different points in its trajectory. We can break down the problem into two separate parts:
Part 1: Motion of the basketball up to the highest point in its trajectory.
Part 2: Motion of the basketball after hitting the rim and bouncing back up.
(a) What is the acceleration of the basketball at the highest point in its trajectory?
To find the acceleration at the highest point, we need to consider the forces acting on the basketball. At the highest point, the only force acting on the basketball is gravity, pulling it downwards. We can assume air resistance is negligible.
The acceleration due to gravity is a constant value of approximately 9.8 m/s², directed downwards. The acceleration at the highest point is therefore equal to the acceleration due to gravity, but in the opposite direction since it is directed upwards. Hence, the acceleration of the basketball at the highest point is -9.8 m/s².
(b) What height above the floor does the basketball reach on this bounce?
To determine the height the basketball reaches after bouncing, we need to use the principle of conservation of mechanical energy. The mechanical energy of the basketball is conserved as long as there are no external forces like air resistance.
The mechanical energy of an object is given by the sum of its kinetic energy (KE) and its potential energy (PE). The kinetic energy depends on the square of the velocity, while the potential energy depends on the height.
1. Initially, before hitting the rim:
The basketball has kinetic energy and potential energy.
KE = 0.5 * m * v^2 (where m is the mass of the basketball, and v is the velocity before hitting the rim)
PE = m * g * h (where g is the acceleration due to gravity, and h is the initial height above the floor)
2. On the way up after bouncing:
The basketball reaches its maximum height, where it briefly comes to rest.
At this point, the velocity is zero, so the kinetic energy is zero.
The PE is now at its maximum since the ball is at its highest point.
3. On the way down after bouncing:
The ball returns to the rim, where it bounces.
The KE is now non-zero, but it is only one-third of the initial KE since the ball rebounds with one-third the initial speed.
To find the height the ball reaches on the bounce, we can equate the initial total mechanical energy (KE + PE) to the final total mechanical energy after the bounce.
Initial energy = Final energy
(0.5 * m * v^2) + (m * g * h) = (0.5 * m * (v/3)^2) + (m * g * h') (where h' is the height above the floor after bouncing)
Since we know the initial height (h) and the initial velocity (v), we can solve this equation to find the height after bouncing (h').
Let's substitute the given values and solve the equation:
(0.5 * m * (10.8 m/s)^2) + (m * 9.8 m/s^2 * 3.05 m) = (0.5 * m * (10.8 m/s / 3)^2) + (m * 9.8 m/s^2 * h')
Simplifying the equation, we can solve for h':
(0.5 * (10.8 m/s)^2) + (9.8 m/s^2 * 3.05 m) = (0.5 * (10.8 m/s / 3)^2) + (9.8 m/s^2 * h')
Solve for h' to find the height above the floor that the basketball reaches on this bounce.